On arc fibers of morphisms of schemes

  • Christopher Chiu

    Eindhoven University of Technology, Eindhoven, Netherlands
  • Tommaso de Fernex

    University of Utah, Salt Lake City, USA
  • Roi Docampo

    University of Oklahoma, Norman, USA
On arc fibers of morphisms of schemes cover
Download PDF

A subscription is required to access this article.

Abstract

Given a morphism of schemes over a field, we prove several finiteness results about the fibers of the induced map on arc spaces. Assuming that is quasi-finite and is separated and quasi-compact, our theorem states that has topologically finite fibers of bounded cardinality and its restriction to , where is the ramification locus of , has scheme-theoretically finite reduced fibers. We also provide an effective bound on the cardinality of the fibers of when is a finite morphism of varieties over an algebraically closed field, describe the ramification locus of , and prove a general criterion for to be a morphism of finite type. We apply these results to further explore the local structure of arc spaces. One application is that the local ring at a stable point of the arc space of a variety has finitely generated maximal ideal and topologically Noetherian spectrum, something that should be contrasted with the fact that these rings are not Noetherian in general; a lower bound on the dimension of these rings is also obtained. Another application gives a semicontinuity property for the embedding dimension and embedding codimension of arc spaces which extends to this setting a theorem of Lech on Noetherian local rings and translates into a semicontinuity property for Mather log discrepancies. Other applications are also discussed.

Cite this article

Christopher Chiu, Tommaso de Fernex, Roi Docampo, On arc fibers of morphisms of schemes. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1500